Stability of Membrane Elastodynamics with Applications to Cylindrical Aneurysms

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="906475.fig.003a" src="https://static.hindawi.com/articles/jam/volume-2011/906475/figures/906475.fig.003a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>Inviscid CSF</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="906475.fig.003b" src="https://static.hindawi.com/articles/jam/volume-2011/906475/figures/906475.fig.003b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b>Viscous CSF</td></tr></table></td></tr><tr class="fig-image" id="c"><td><img alt="906475.fig.003c" src="https://static.hindawi.com/articles/jam/volume-2011/906475/figures/906475.fig.003c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b>Viscous CSF, <svg height="18.1" id="M323" style="vertical-align:-5.76988pt" version="1.1" viewbox="0 0 58.049999 18.1" width="58.049999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.863)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g>
<g transform="matrix(.012,-0,0,-.012,13.987,14.938)"><path d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z" id="x1D453"></path></g>
<g transform="matrix(.017,-0,0,-.017,26.962,10.863)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,41.666,10.863)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,49.826,10.863)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g>
</svg></td></tr></table></td></tr><tr class="fig-image" id="d"><td><img alt="906475.fig.003d" src="https://static.hindawi.com/articles/jam/volume-2011/906475/figures/906475.fig.003d.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(d) </b>Viscoelastic model</td></tr></table></td></tr></table>

<table>Phase portraits for inviscid CSF, viscous CSF, and viscoelastic model.</table>

Journal of Applied Mathematics

fig3

Figure 3

Figure 3: Stability of Membrane Elastodynamics with Applications to Cylindrical Aneurysms 